LMS Invited Lecturer Fedor Bogomolov

Birational Geometry and Galois Groups

I am going to discuss the relation between the structure of the Galois group of algebraic closure of a field of rational functions and the structure of the field itself. More precisely how to extract effectively birational invariants (i.e. geometric invariants of projective models of the functional field) from the structure of the Galois group of its algebraic closure.

Thus in first the lectures I will discuss the notion of stable (the same as essential) cohomology for groups and varieties. In particular I will consider the notion of unramified cohomology for groups and algebraic varieties which provide natural birational invariants of the varieties.

The solution of the Bloch-Kato conjecture by Voevodsky shows that there is a particular quotient of the Galois group in the functional case containing almost the information about the field of function.

I will discuss the structure of the group which is the maximal pro-l-quotient of . It is a central pro-l-extension of the maximal abelian l-quotient of the Galois group of the algebraic closure of the functional field.

My goal is to provide a sketch of the proof of the theorem (BT) which states that (functorially) defines $K$ modulo purely inseparble extension if K has transcendence degree bigger or equal than 2 over the algebraically closed field k. In fact I will outline a proof of a result that with a fan of subgroups with abelian preimages in (liftable subgroups) defines K modulo purely inseparable extensions. It implies that (most of) finite birational invariants can be modelled on finite abelian groups with a fan of subgroups.

I will also discuss some applications of the above results which include the theory of algebraic universal spaces for finite birational invariants, section conjecture in the functional case. I also plan to discuss so called "freedom comnjecture" and some other related questions in birational geometry.

My last topic will concern some open questions in algebraic geometry and in particular in the the theory of rational and unirational varieties.

The lectures cover the results obtained mostly in collaboration with Yuri Tschinkel and Christian Bohning.

Support Lectures

Fano four-folds (Brown)

Abstract: The cubic surface (smooth, projective) and its 27 lines is one starting point of higher-dimensional birational geometry. More generally, the class of del Pezzo surfaces, surfaces with ample anticanonical class, is a major corner of the classification of surfaces, and appears widely throughout algebraic and arithmetic geometry. In 3-dimensions, the analogous class is Fano 3-folds. Nonsingular Fano 3-folds are classified very famously by Fano, Iskovskikh, Mori, Mukai and others. Even allowing singularities of various classes, it is known that there are only finitely many families of Fano 3-folds, and there is a lot of work towards a classification, even though it seems a long way off at the moment.

Fano 4-folds are also known to arise in only finitely many families, although far less is known about these. I will explain the formal definitions, say a little about the classes of singularities one might consider, mention the role of orbifolds, and explain how to extend the work on hypersurfaces of Johnson--Koll\'ar and Reid to classify Fano orbifolds in 4 dimensions.

Going beyond hypersurfaces, I will show examples of Fano 4-folds that are not hypersurfaces. It is interesting also to consider Calabi--Yau 3-folds that may be members of the anticanonical system of a Fano 4-fold (in the same way as one considers K3 surfaces that may lie on a Fano 3-fold) and I show how these may be constructed in higher codimension using the classical idea of inverting projections.

Subgroups of Cremona groups (Cheltsov)

Abstract: I will explain how to study conjugacy classes of finite subgroups in Cremona groups.

Birational Geometry and Derived Categories (Logvinenko)

Abstract: These talks give an introduction to working with the bounded derived category of coherent sheaves on an algebraic variety as a useful tool for studying its geometry. We review the main definitions and then proceed to look at the known applications in the field of birational geometry.

Invited talks

Birational geometry of moduli of sheaves on K3s via wall-crossing. (Bayer)

Abstract: I will explain joint work with Emanuele Macrž, in which we systematically use wall-crossing for Bridgeland stability conditions to study the birational geometry of moduli spaces of stable sheaves on K3 surfaces. Our results include a description of the nef cone in terms of the lattice of the K3 surface, and a proof of a well-known conjecture on the existence of Lagrangian fibrations. These results are new even in the case of the Hilbert scheme. They yield many concrete applications in examples.

The F-splitting ratio of a toric variety. (Hering)

Abstract: The Frobenius morphism is a useful tool in the study of algebraic varieties. One of its uses is to give a measurement of how bad the singularities of a ring are. This measurement is called the the F-splitting ratio, which agrees with the F-signature for normal rings. The F-signature of a normal toric ring was computed by Von Korff. I will give give an introduction to these notions and present the computation of the F-splitting ratio of a seminormal ring. This is joint work with Kevin Tucker.

0-cycles on intersections of quadrics and cubics in P^4, Prymians and relevant motivic stuff (Guletskii)

One exercise in (stable) birational geometry (Ilya Karzhemanov)

Abstract: I will talk about on how the existence of rational fibrations by abelian hypersurfaces on a variety X serve as an obstruction for X to be (stably) rational. The exercise in question is to show that X = P^d carries this type of fibrations (any d>1). The case d=2 is classical (and one recovers the notion of a Halphen pencil here). I will describe a way to attack the d>2 case.

On stable conjugacy of finite subgroups of the plane Cremona group. (Yuri Prokhorov)

Abstract: I am going to discuss the problem of stable conjugacy of finite subgroups of Cremona groups. In particular, I will prove that in dimension two, except for a few cases, stable birational triviality implies birational triviality.

New invariants of contractible curves and applications (Wemyss)

Abstract: I will describe how to attach to every contractible curve in a 3-fold a (in general non-commutative) algebra called the contraction algebra. I will explaine how to recover Reid's width of a curve as a special case, and also how to obtain the commutative deformation base as a certain quotient. As an application I will show how to describe Bridgeland's flop-flop functor in terms of the contraction algebra, and describe why previous approaches fail. This is based on joint work with Will Donovan.